Exercises 3.5 Exercises
In the following exercises find a basis and the dimension of vector space.
1.
\(\C\) over \(\R\text{.}\)2.
\(F^n\) over a field \(F\) (refer to Example 2.2.2)3.
\(\End_{\R}(\R)\) over \(\R\) (refer to Example 2.2.3).4.
\(M_{m\times n}(F)\) over a field \(F\text{.}\)5.
Vector space of all polynomials in one variable of degree at most \(n\) over a field \(F\text{.}\)In the following exercises assume that \(V\) is a finite-dimensional vector space over a field \(F\text{.}\)
6.
Let \(\{v_1,v_2,\ldots,v_n\}\) be a basis of \(V\text{.}\) Show that vectors
\begin{equation*}
w_i=\sum_{j=1}^{n}\alpha_{ij}v_j \quad \text{for } 1\leq i\leq m
\end{equation*}
are linearly dependent if and only if
\begin{equation*}
\begin{pmatrix}\alpha_{11}\amp\alpha_{21}\amp\cdots\amp\alpha_{m1}\\
\alpha_{12}\amp\alpha_{22}\amp\cdots\amp\alpha_{m2}\\
\vdots\amp\vdots\amp\ddots\amp\vdots\\
\alpha_{1n}\amp\alpha_{2n}\amp\cdots\amp\alpha_{mn}\\\end{pmatrix}X=0
\end{equation*}
has a nontrivial solution.